# Fast and Provable ADMM for Learning with Generative Priors

arXiv: https://arxiv.org/abs/1907.03343 · HTML: https://mlsanity.com/paper/1907.03343/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jul 2019.

## Built on by (most cited first, 18 of 18)

- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Subquadratic Overparameterization for Shallow Neural Networks](https://mlsanity.com/paper/2111.01875/) (2021)
- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition](https://mlsanity.com/paper/2106.03668/) (2021)
- [Theoretical Perspectives on Deep Learning Methods in Inverse Problems](https://mlsanity.com/paper/2206.14373/) (2022)
- [Prior Image-Constrained Reconstruction using Style-Based Generative Models](https://mlsanity.com/paper/2102.12525/) (2021)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [TurbuGAN: An Adversarial Learning Approach to Spatially-Varying Multiframe Blind Deconvolution with Applications to Imaging Through Turbulence](https://mlsanity.com/paper/2203.06764/) (2022)
- [Inverse Problems Leveraging Pre-trained Contrastive Representations](https://mlsanity.com/paper/2110.07439/) (2021)
- [Inverse problem regularization with hierarchical variational autoencoders](https://mlsanity.com/paper/2303.11217/) (2023)
- [Convergent Data-driven Regularizations for CT Reconstruction](https://mlsanity.com/paper/2212.07786/) (2022)
- [On the Fine-Grained Hardness of Inverting Generative Models](https://mlsanity.com/paper/2309.05795/) (2023)
- [VDIP-TGV: Blind Image Deconvolution via Variational Deep Image Prior Empowered by Total Generalized Variation](https://mlsanity.com/paper/2310.19477/) (2023)
- [Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview](https://mlsanity.com/paper/2402.12072/) (2024)
- [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026)
- [The Nonconvex Geometry of Linear Inverse Problems](https://mlsanity.com/paper/2101.02776/) (2021)
- [Coefficient Shape Alignment in Multiple Functional Linear Regression](https://mlsanity.com/paper/2312.01925/) (2023)

## Abstract (arXiv)

In this work, we propose a (linearized) Alternating Direction Method-of-Multipliers (ADMM) algorithm for minimizing a convex function subject to a nonconvex constraint. We focus on the special case where such constraint arises from the specification that a variable should lie in the range of a neural network. This is motivated by recent successful applications of Generative Adversarial Networks (GANs) in tasks like compressive sensing, denoising and robustness against adversarial examples. The derived rates for our algorithm are characterized in terms of certain geometric properties of the generator network, which we show hold for feedforward architectures, under mild assumptions. Unlike gradient descent (GD), it can efficiently handle non-smooth objectives as well as exploit efficient partial minimization procedures, thus being faster in many practical scenarios.